IMAT Physics · Heat and Thermodynamics

Specific Heat, Latent Heat and Mixing

Heating a body raises its temperature by an amount set by its mass and specific heat; at a change of state the energy goes into the change and the temperature stays put.

All Heat and Thermodynamics notes

Why this matters

Six of the 13 past questions are on this page: rearranging Q = mcΔT, a heater's power or a latent heat found from a heater, what happens to a solid as it melts, and in 2023 a ministry question on ice melting in warm water.

Concept 1 of 3: Specific heat capacity: the energy to warm a mass

Twice the mass needs twice the energy for the same warming, and twice the warming needs twice the energy. The material matters too: water needs about ten times more energy per kilogram than iron for the same rise. The specific heat capacity is that material constant.

Definition

The specific heat capacity $c$ of a material is the energy needed to raise the temperature of 1 kg of it by 1 K. Unit: $\text{J kg}^{-1}\text{K}^{-1}$ .
Water has a very high value, about $4200\ \text{J kg}^{-1}\text{K}^{-1}$ ; most metals are a few hundred.
The heat capacity of a whole object is $C = mc$ , the energy per kelvin for that object.
An electric heater of power $P$ running for time $t$ supplies $Q = Pt$ , with $t$ in seconds.

Heating without a change of state

Q = m c \Delta T \qquad Q = P t

$Q$ energy supplied, in J
$m$ mass, in kg
$c$ specific heat capacity, in J kg⁻¹ K⁻¹
$\Delta T$ temperature change, in K
$P,\ t$ heater power in W, and time in s

Worked example

How much energy is needed to heat 0.50 kg of water from 20 °C to 100 °C? Take

c = 4200\ \text{J kg}^{-1}\text{K}^{-1}

. How long does a 2.0 kW kettle take to supply it, if no energy is lost?

Practice this conceptself-check · 4 quick reps

Power times time needs seconds

A watt is a joule per second, so

Q = Pt

gives joules only with

t

in seconds. Leaving minutes in gives an answer 60 times too small, and that wrong value is usually one of the options.

Thermal conductivity is not specific heat capacity

Specific heat capacity says how much energy warms a kilogram; thermal conductivity says how fast heat passes through a material. A calorimetry calculation never needs the conductivity.

Concept 2 of 3: Latent heat and changes of state

Keep heating melting ice and the thermometer stays at 0 °C until all the ice has gone. The energy is not lost: it is pulling the molecules apart against their attractions, so their potential energy rises while their kinetic energy, and so the temperature, does not. Boiling takes far more energy than melting, because the molecules must be separated completely.

Definition

The specific latent heat $L$ is the energy needed to change the state of 1 kg of a substance at constant temperature. Unit: $\text{J kg}^{-1}$ .
Fusion ( $L_f$ ) is solid to liquid; vaporisation ( $L_v$ ) is liquid to gas. For water, $L_f \approx 3.3 \times 10^5\ \text{J kg}^{-1}$ and $L_v \approx 2.3 \times 10^6\ \text{J kg}^{-1}$ , about seven times more.
During a change of state the temperature stays constant: the energy overcomes the forces between molecules. Bonds inside the molecules are not broken.
Melting and boiling make the particles more disordered. Freezing and condensing release the same latent heat back.
Evaporation happens at the surface at any temperature; the fastest molecules escape, so the liquid left behind cools (why sweating cools you). Boiling happens throughout the liquid at one fixed temperature.

Energy for a change of state

Q = m L

$Q$ energy absorbed or released, in J
$m$ mass that changes state, in kg
$L$ specific latent heat, in J kg⁻¹

Worked example

Take

L_f = 3.34 \times 10^5\ \text{J kg}^{-1}

and

L_v = 2.26 \times 10^6\ \text{J kg}^{-1}

for water. How much energy melts 0.20 kg of ice at 0 °C, and how much boils away 0.20 kg of water at 100 °C?

Practice this conceptself-check · 4 quick reps

Melting breaks forces between molecules, not bonds inside them

When wax or ice melts, the energy overcomes the attractions between molecules. The covalent bonds within each molecule are untouched; breaking those would be a chemical change. Statements that melting breaks the bonds inside the molecules are wrong.

The temperature does not rise during a change of state

While a pure substance melts or boils, the energy supplied goes into the change of state and the temperature stays constant. A heating curve shows a flat section there, not a slope.

Concept 3 of 3: Calorimetry: heat lost equals heat gained

In an insulated container no energy escapes, so every joule the hot things lose is gained by the cold things. Write one side for each, set them equal, and solve. If ice melts, the cold side has two terms: the energy to melt it, and then the energy to warm the meltwater up from 0 °C.

Definition

In an insulated system, energy lost by the hotter bodies = energy gained by the colder bodies, and all end at one final temperature $T$ .
Write each change as a positive amount: (higher temperature minus lower temperature).
If ice at 0 °C melts completely, the gain is $m_{\text{ice}} L_f + m_{\text{ice}} c_w (T - 0)$ . Leaving out the second term is the classic slip.
The final temperature is the simple average only if the two masses and their specific heats are equal.

Mixing in an insulated container

m_1 c_1 (T_1 - T) = m_2 c_2 (T - T_2)

$m_1, c_1, T_1$ mass, specific heat and starting temperature of the hotter body
$m_2, c_2, T_2$ the same for the colder body
$T$ the shared final temperature

Worked example

0.10 kg of ice at 0 °C is dropped into 0.90 kg of water at 40 °C in an insulated flask. All the ice melts. Take

c_w = 4200\ \text{J kg}^{-1}\text{K}^{-1}

and

L_f = 3.36 \times 10^5\ \text{J kg}^{-1}

. Find the final temperature.

Practice this conceptself-check · 3 quick reps

Melted ice must still warm up

When ice melts and the mixture ends above 0 °C, the meltwater also warms from 0 °C to the final temperature. The energy balance needs

m L_f + m c T

on the gaining side, not

m L_f

alone.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Specific heat capacity: the energy to warm a mass
Heating without a change of state
$Q = m c \Delta T \qquad Q = P t$
Latent heat and changes of state
Energy for a change of state
$Q = m L$
Calorimetry: heat lost equals heat gained
Mixing in an insulated container
$m_1 c_1 (T_1 - T) = m_2 c_2 (T - T_2)$

Watch out for (5)

Power times time needs seconds
→ Specific heat capacity: the energy to warm a mass
Thermal conductivity is not specific heat capacity
→ Specific heat capacity: the energy to warm a mass
Melting breaks forces between molecules, not bonds inside them
→ Latent heat and changes of state
The temperature does not rise during a change of state
→ Latent heat and changes of state
Melted ice must still warm up
→ Calorimetry: heat lost equals heat gained

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